Finite Group Databasebeta

← #35462644

Subgroups — #35462644

Order 21248.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912484C211yesmaximal, characteristic, complemented
2#24908851C211yesmaximal, characteristic
3#24908851C211yesmaximal, characteristic
4#24908609C211yesmaximal, characteristic
5#24908599C211yesmaximal, characteristic
6#24908420C211yesmaximal, characteristic, complemented
7#24908411C211yesmaximal, characteristic, complemented
8#24908219C211yesmaximal, characteristic, complemented
9#24908187C211yesmaximal, characteristic, complemented
10#24908055C211yesmaximal, characteristic, complemented
11#24908034C211yesmaximal, characteristic, complemented
12#24907598C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2354C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581703C2211yescharacteristic
28#19127634—21no—
29#19127621—21no—
30#19127605—21no—
31#19127599C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127598—21no—
34#19127567C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C4×D4×C83C2211yescharacteristic
49C2×#117×C83—21no—
50#390×C83C2211yescharacteristic
51#390×C83C2211yescharacteristic
52#388×C83—21no—
53#388×C83—21no—
54#384×C83C2211yescharacteristic
55#380×C83—21no—
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C22×Dic166—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581792—22no—
65#14581792—41no—
66#14581784—22no—
67#14581784—41no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C4×Dic83—21no—
78#14581725—22no—
79#14581725—22no—
80C4⋊(C83⋊C8)—22no—
81C4⋊(C83⋊C8)—41no—
82C4⋊(C83⋊C8)—41no—
83#14581713—22no—
84#14581713—22no—
85#14581713—41no—
86#14581713—41no—
87#14581703—22no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C22×D4×C83C2311yescharacteristic
94C23×C4×C83—21no—
95C4×D4×C83—22no—
96C4×D4×C83Q814yescomplemented
97#117×C83—44no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83D411yescharacteristic
101C2×C4⋊C4×C83D411yescharacteristic
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83—21no—
109C2×C22⋊C4×C83—21no—
110C2×C42×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83C2311yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115#95×C83—21no—
116#95×C83—21no—
117#95×C83D411yescharacteristic
118#95×C83D411yescharacteristic
119C2×Dic166—42no—
120C2×Dic166—42no—
121C2×Dic166—81no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—42no—
145C4⋊Dic83#4112yes—
146C4⋊Dic83#4112yes—
147Dic83⋊C4—42no—
148Dic83⋊C4—42no—
149Dic83⋊C4—42no—
150Dic83⋊C4—81no—
151C4×Dic83—81no—
152C24×C83—21no—
153C2×D4×C83—21no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83C2×Q812yes—
159C22×C4×C83—21no—
160C22×C4×C83—24no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—42no—
164C22×C4×C83—42no—
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2411yescharacteristic
172C4⋊C4×C83—22no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—41no—
179C4⋊C4×C83—42no—
180C4⋊C4×C83—81no—
181C4⋊C4×C83C2×Q812yes—
182C22⋊C4×C83—24no—
183C22⋊C4×C83—42no—
184C22⋊C4×C83—42no—
185C22⋊C4×C83—81no—
186C22⋊C4×C83—81no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—42no—
192C42×C83—42no—
193C42×C83C2×Q812yes—
194C42×C83#4112yes—
195C23×C83—22no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83C22×D411yescharacteristic
203D4×C83—22no—
204D4×C83—22no—
205D4×C83—41no—
206D4×C83—41no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—22no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—48no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83C22⋊Q812yes—
233C2×C4×C83C22⋊Q812yes—
234C2×C4×C83AGL(1,ℤ/8)12yes—
235C2×C4×C83C22×Q811yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83H(5,𝔽2)11yescharacteristic
240C2×C4×C832−1+411yescharacteristic
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253Dic166—84no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C83⋊C8—41no—
257C83⋊C8—41no—
258C22×C83—42no—
259C22×C83—42no—
260C22×C83—42no—
261C22×C83—81no—
262C22×C83—81no—
263C22×C83—81no—
264C22×C83—81no—
265C22×C83C2×C22⋊Q811yescharacteristic
266C22×C83#54611yescharacteristic
267C22×C83#54811yescharacteristic
268C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C83#57511yescharacteristic
270C22×C83#57611yescharacteristic
271C22×C83#57811yescharacteristic
272C4×C83—22no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—42no—
276C4×C83—41no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—81no—
280C4×C83—81no—
281C4×C83#53311yescharacteristic
282C4×C83#54411yescharacteristic
283C4×C83#55511yescharacteristic
284C4×C83#55611yescharacteristic
285Dic83—42no—
286Dic83—42no—
287Dic83—81no—
288Dic83—81no—
289#26033—831nomaximal, Sylow
290C2×C83—82no—
291C2×C83#243911yescharacteristic
292C2×C83#307711yescharacteristic
293C2×C83#308311yescharacteristic
294C2×C83#330811yescharacteristic
295C2×C83#331911yescharacteristic
296C2×C83#336611yescharacteristic
297C2×C83#337711yescharacteristic
298#2550—831no—
299#2354D8311yescharacteristic
300#2067—831no—
301#2067—831no—
302#2025—831no—
303#2023—831no—
304#1989—831no—
305#1987—831no—
306#1947—831no—
307#1940—831no—
308#1921—831no—
309#1918—831no—
310#1836—831no—
311#1804—831no—
312#1762—831no—
313C83#2603311yescharacteristic, complemented, Sylow
314C2×C4⋊Q8—831no—
315C2×#125—831no—
316C2×C4×D4C2×D8311yescharacteristic
317C2×#117—21no—
318C2×#107—831no—
319C2×#107—831no—
320C2×C4⋊C8—831no—
321C2×C4⋊C8—831no—
322C2×C4⋊C8—831no—
323C2×#102—831no—
324C2×#102—831no—
325C2×#102—831no—
326C2×#102—831no—
327C2×#97—831no—
328#398—1661no—
329#395—1661no—
330#391—1661no—
331#390C2×D8311yescharacteristic
332#390C2×D8311yescharacteristic
333#389—831no—
334#389—831no—
335#389—1661no—
336#388—21no—
337#388—21no—
338#384C2×D8311yescharacteristic
339#382—831no—
340#380—21no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4C2×D8311yescharacteristic
343C4×C4⋊C4C2×D8311yescharacteristic
344C4×C4⋊C4C2×D8311yescharacteristic
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#336—831no—
356#336—831no—
357C22×Q8—1661no—
358C22×D4C22×D8311yescharacteristic
359C23×C4—21no—
360C22×C8—831no—
361C22×C8—831no—
362C22×C8—831no—
363C22×C8—831no—
364C4⋊Q8—3321no—
365C4⋊Q8—1662no—
366#125—3321no—
367#125—1662no—
368C4×D4—22no—
369C4×D4Dic16614yescomplemented
370#117—44no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4C22⋊D8311yescharacteristic
383C2×C4⋊C4C22⋊D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—1661no—
395C2×C42C22×D8311yescharacteristic
396C2×C42C22×D8311yescharacteristic
397C2×C42C22×D8311yescharacteristic
398#107—1662no—
399#107—1662no—
400C4⋊C8—3321no—
401C4⋊C8—3321no—
402C4⋊C8—1662no—
403#102—3321no—
404#102—3321no—
405#102—1662no—
406#102—1662no—
407#97—1662no—
408#95—21no—
409#95—21no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95C22⋊D8311yescharacteristic
416#95C22⋊D8311yescharacteristic
417C24—21no—
418C2×Q8—3322no—
419C2×Q8—3322no—
420C2×Q8—6641no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Dic16612yes—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C23×D8311yescharacteristic
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—3322no—
466C4⋊C4—3322no—
467C4⋊C4—3322no—
468C4⋊C4—3322no—
469C4⋊C4—832no—
470C4⋊C4—832no—
471C4⋊C4—6641no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4C2×Dic16612yes—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—42no—
490C42—6641no—
491C42C2×Dic16612yes—
492C42#1305059012yes—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C22×C22⋊D8311yescharacteristic
501Q8—6644no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4#1458177112yes—
544C2×C4#1458177112yes—
545C2×C4#1458181812yes—
546C2×C4C22×Dic16611yescharacteristic
547C2×C4#41×D8311yescharacteristic
548C2×C4#41×D8311yescharacteristic
549C2×C4C2×#1305059011yescharacteristic
550C2×C4#1458183811yescharacteristic
551C2×C4#1458183911yescharacteristic
552C8—3321no—
553C8—3321no—
554C8—3321no—
555C8—3321no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C2×#1458177111yescharacteristic
564C22#1912833911yescharacteristic
565C22#1912835811yescharacteristic
566C22C2×#1458181811yescharacteristic
567C22#1912849911yescharacteristic
568C22#1912850411yescharacteristic
569C22#1912851611yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—3322no—
580C4—3322no—
581C4—6641no—
582C4—6641no—
583C4#1912825411yescharacteristic
584C4#1912832911yescharacteristic
585C4#1912841411yescharacteristic
586C4#1912841911yescharacteristic
587C2—82no—
588C2#2491114811yescharacteristic
589C2#2491726011yescharacteristic
590C2#2491731411yescharacteristic
591C2#2491944811yescharacteristic
592C2#2491961311yescharacteristic
593C2#2492016611yescharacteristic
594C2#2492029911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2354C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2354D83
hypocenterC83#26033
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #24912484C2semidirect
6. #24908420C2semidirect
7. #24908411C2semidirect
8. #24908219C2semidirect
9. #24908187C2semidirect
10. #24908055C2semidirect
11. #24908034C2semidirect
12. #24907598C2semidirect
96. C4×D4×C83Q8semidirect
313. C83#26033semidirect
369. C4×D4Dic166semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#249124842
#249088512
#249086092
#249085992
#249084202
#249084112
#249082192
#249081872
#249080552
#249080342
#249075982
#249074732
#249072882
C83×#23542
#2603383